let s be State of ; :: thesis:
set D = Int-Locations \/ FinSeq-Locations ;
let I, J be Program of ; :: thesis:
let a be read-write Int-Location ; :: thesis:
set J0 = Directed J;
set s0 = Initialize s;
set J9 = J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )));
set s00 = s +* ((Directed J) +* (Start-At (insloc 0 )));
set s3 = s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )));
set s4 = Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1;
set s5 = Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2;
set i = a =0_goto (insloc ((card J) + 3));
A1: (Directed J) +* (Start-At (insloc 0 )) c= s +* ((Directed J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
if=0 a,I,J = ((((a =0_goto (insloc ((card J) + 3))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' I) ';' (Stop SCM+FSA ) by SCMFSA8B:def 1;
then if=0 a,I,J = (((Macro (a =0_goto (insloc ((card J) + 3)))) ';' J) ';' (Goto (insloc ((card I) + 1)))) ';' (I ';' (Stop SCM+FSA )) by SCMFSA6A:67;
then if=0 a,I,J = ((Macro (a =0_goto (insloc ((card J) + 3)))) ';' J) ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA ))) by SCMFSA6A:67;
then A2: if=0 a,I,J = (Macro (a =0_goto (insloc ((card J) + 3)))) ';' (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))) by SCMFSA6A:67;
card (Macro (a =0_goto (insloc ((card J) + 3)))) = 2 by SCMFSA7B:6;
then A3: ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) c= if=0 a,I,J by A2, FUNCT_4:26;
A4: insloc 0 in dom (if=0 a,I,J) by Th54;
if=0 a,I,J c= (if=0 a,I,J) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then dom (if=0 a,I,J) c= dom ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
then A5: (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 0 ) = ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (insloc 0 ) by A4, FUNCT_4:14
.= (if=0 a,I,J) . (insloc 0 ) by A4, SCMFSA6B:7
.= a =0_goto (insloc ((card J) + 3)) by Th55 ;
card (if=0 a,I,J) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:14
.= ((card J) + 2) + ((card I) + 2) ;
then A6: ((card J) + 2) + 0 < card (if=0 a,I,J) by XREAL_1:10;
then A7: insloc ((card J) + 2) in dom (if=0 a,I,J) by SCMFSA6A:15;
IC SCM+FSA in dom ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by SF_MASTR:65;
then A8: IC (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by FUNCT_4:14
.= insloc 0 by SF_MASTR:66 ;
set ss = Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2);
A9: if=0 a,I,J c= (if=0 a,I,J) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
A10: if=0 a,I,J c= s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26, SCMFSA6B:5;
(if=0 a,I,J) +* (Start-At (insloc 0 )) c= s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26;
then if=0 a,I,J c= s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by A9, XBOOLE_1:1;
then ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) c= s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by A3, XBOOLE_1:1;
then ProgramPart [(ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2))] c= Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by AMI_1:99;
then A11: ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) c= Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by AMI_1:105;
ProgramPart (Relocated (Directed J),2) c= ProgramPart (Relocated (J ';' ((Goto (insloc ((card I) + 1))) ';' (I ';' (Stop SCM+FSA )))),2) by Th12, SCMFSA6A:55;
then A12: ProgramPart (Relocated (Directed J),2) c= Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2 by A11, XBOOLE_1:1;
card (if=0 a,I,J) = ((card I) + (card J)) + (3 + 1) by SCMFSA8B:14
.= (((card I) + (card J)) + 3) + 1 ;
then ((card I) + (card J)) + 3 < card (if=0 a,I,J) by NAT_1:13;
then A13: insloc (((card I) + (card J)) + 3) in dom (if=0 a,I,J) by SCMFSA6A:15;
assume s . a <> 0 ; :: thesis:
then A14: (Initialize s) . a <> 0 by SCMFSA6C:3;
A15: insloc 1 in dom (if=0 a,I,J) by Th54;
if=0 a,I,J c= (if=0 a,I,J) +* (Start-At (insloc 0 )) by SCMFSA8A:9;
then A16: dom (if=0 a,I,J) c= dom ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
A17: if=0 a,I,J c= s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by FUNCT_4:26, SCMFSA6B:5;
assume A18: Directed J is_pseudo-closed_on s ; :: thesis:
then A19: pseudo-LifeSpan s,(Directed J) = LifeSpan (s +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 )))) by Th58;
A20: (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) . (insloc 1) = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 1) by AMI_1:54
.= ((if=0 a,I,J) +* (Start-At (insloc 0 ))) . (insloc 1) by A15, A16, FUNCT_4:14
.= (if=0 a,I,J) . (insloc 1) by A15, SCMFSA6B:7
.= goto (insloc 2) by Th55 ;
DataPart s = DataPart (s +* ((Directed J) +* (Start-At (insloc 0 )))) by SCMFSA8A:11;
then A21: Directed J is_pseudo-closed_on s +* ((Directed J) +* (Start-At (insloc 0 ))) by A18, Th52;
A22: Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(0 + 1) = Following (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),0 ) by AMI_1:14
.= Following (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by AMI_1:13
.= Exec (a =0_goto (insloc ((card J) + 3))),(s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by A8, A5 ;
(s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . a = s . a by Th28
.= (Initialize s) . a by SCMFSA6C:3 ;
then A23: IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) = Next (IC (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))))) by A14, A22, SCMFSA_2:96
.= insloc (0 + 1) by A8 ;
A24: Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(1 + 1) = Following (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) by AMI_1:14
.= Exec (goto (insloc 2)),(Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) by A23, A20 ;
then A25: IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) = insloc 2 by SCMFSA_2:95;
A26: DataPart (s +* ((Directed J) +* (Start-At (insloc 0 )))) = DataPart (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) by SCMFSA6A:39, SCMFSA8A:14;

A27: now

let f be FinSeq-Location ; :: thesis:
thus (s +* ((Directed J) +* (Start-At (insloc 0 )))) . f = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . f by A26, SCMFSA6A:38
.= (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) . f by A22, SCMFSA_2:96
.= (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) . f by A24, SCMFSA_2:95 ; :: thesis:

end;

now

let a be Int-Location ; :: thesis:
thus (s +* ((Directed J) +* (Start-At (insloc 0 )))) . a = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . a by A26, SCMFSA6A:38
.= (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),1) . a by A22, SCMFSA_2:96
.= (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) . a by A24, SCMFSA_2:95 ; :: thesis:

end;

then A28: DataPart (s +* ((Directed J) +* (Start-At (insloc 0 )))) = DataPart (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) by A27, SCMFSA6A:38;
A29: IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2)) = IC (Computation (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2),(pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J))) by AMI_1:51
.= (IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)))) + 2 by A1, A21, A12, A25, A28, Th51
.= (IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),(pseudo-LifeSpan s,(Directed J)))) + 2 by A18, Th50
.= (insloc (card (ProgramPart (Directed J)))) + 2 by A18, SCMFSA8A:def 5
.= (insloc (card (Directed J))) + 2 by AMI_1:105
.= insloc ((card J) + 2) by SCMFSA8A:34 ;
then A30: CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2)) = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc ((card J) + 2)) by AMI_1:54
.= (if=0 a,I,J) . (insloc ((card J) + 2)) by A10, A7, GRFUNC_1:8
.= goto (insloc (((card I) + (card J)) + 3)) by Th63 ;
A31: IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2) + 1)) = IC (Following (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2))) by AMI_1:14
.= insloc (((card I) + (card J)) + 3) by A30, SCMFSA_2:95 ;
then A32: CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),(((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2) + 1)) = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc (((card I) + (card J)) + 3)) by AMI_1:54
.= (if=0 a,I,J) . (insloc (((card I) + (card J)) + 3)) by A17, A13, GRFUNC_1:8
.= halt SCM+FSA by Th61 ;
then A33: ProgramPart (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) halts_on s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by AMI_1:146;
hence if=0 a,I,J is_halting_on s by SCMFSA7B:def 8; :: thesis:

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose k = 0 ; :: thesis:

then Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k = s +* ((if=0 a,I,J) +* (Start-At (insloc 0 ))) by AMI_1:13;
then IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = insloc 0 by Th31;
hence IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if=0 a,I,J) by Th54; :: thesis:

end;

suppose ( 0 < k & k = 1 ) ; :: thesis:

hence IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if=0 a,I,J) by A23, Th54; :: thesis:

end;

suppose A34: ( 0 < k & k <> 1 & k < (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2 ) ; :: thesis:

then 0 + 1 <= k by INT_1:20;
then 1 < k by A34, XXREAL_0:1;
then 0 + (1 + 1) <= k by INT_1:20;
then consider k2 being Nat such that
A35: 2 + k2 = k by NAT_1:10;
card (if=0 a,I,J) = ((card I) + (card J)) + (2 + 2) by SCMFSA8B:14
.= ((card J) + 2) + ((card I) + 2) ;
then A36: (card J) + 2 <= card (if=0 a,I,J) by NAT_1:12;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),k2) as Element of NAT by ORDINAL1:def 13;
A37: k2 < pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J) by A34, A35, XREAL_1:8;
then k2 < pseudo-LifeSpan s,(Directed J) by A18, Th50;
then insloc n in dom (Directed J) by A18, SCMFSA8A:31;
then n < card (Directed J) by SCMFSA6A:15;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:8;
then n + 2 < (card J) + 2 by SCMFSA8A:34;
then A38: n + 2 < card (if=0 a,I,J) by A36, XXREAL_0:2;
IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = IC (Computation (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2),k2) by A35, AMI_1:51
.= (IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),k2)) + 2 by A1, A21, A12, A25, A28, A37, Th51 ;
hence IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if=0 a,I,J) by A38, SCMFSA6A:15; :: thesis:

end;

suppose A39: ( 0 < k & k <> 1 & (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2 <= k ) ; :: thesis:

hereby :: thesis:

per cases ) + 2 = k or (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2 < k ) by A39, XXREAL_0:1;

suppose (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2 = k ; :: thesis:

hence IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if=0 a,I,J) by A6, A29, SCMFSA6A:15; :: thesis:

end;

suppose (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2 < k ; :: thesis:

then ((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 2) + 1 <= k by INT_1:20;
hence IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) in dom (if=0 a,I,J) by A13, A31, A32, AMI_1:52; :: thesis:

end;

end;

end;

end;

end;

end;

hence if=0 a,I,J is_closed_on s by SCMFSA7B:def 7; :: thesis:
A40: CurInstr (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = a =0_goto (insloc ((card J) + 3)) by A8, A5;

now

A41: 0 + 2 < ((card I) + (card J)) + 3 by XREAL_1:10;
then A42: insloc 2 in dom (if=0 a,I,J) by Th56;
A43: (if=0 a,I,J) . (insloc 2) <> halt SCM+FSA by A41, Th56;
A44: CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2) = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (insloc 2) by A25, AMI_1:54
.= (if=0 a,I,J) . (insloc 2) by A42, Th26 ;
let k be Element of NAT ; :: thesis:
assume A45: CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = halt SCM+FSA ; :: thesis:
CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),0 ) = a =0_goto (insloc ((card J) + 3)) by A40, AMI_1:13;
then A46: k <> 0 by A45, SCMFSA_2:48, SCMFSA_2:124;
k <> 1 by A23, A20, A45, SCMFSA_2:47, SCMFSA_2:124;
then 2 < k by A45, A46, A43, A44, NAT_1:27;
then consider k2 being Nat such that
A47: 2 + k2 = k by NAT_1:10;
reconsider k2 = k2 as Element of NAT by ORDINAL1:def 13;
reconsider n = IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),k2) as Element of NAT by ORDINAL1:def 13;
assume not (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + (1 + 2) <= k ; :: thesis:
then k < ((pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 1) + 2 ;
then k2 < (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 1 by A47, XREAL_1:8;
then A48: k2 <= pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J) by NAT_1:13;
then A49: k2 <= pseudo-LifeSpan s,(Directed J) by A18, Th50;

A50: now

per cases by A49, XXREAL_0:1;

suppose A51: k2 = pseudo-LifeSpan s,(Directed J) ; :: thesis:

((card I) + (card J)) + (2 + 1) = (((card J) + 2) + 1) + (card I) ;
then A52: ((card J) + 2) + 1 <= ((card I) + (card J)) + 3 by NAT_1:11;
IC (Computation (s +* ((Directed J) +* (Start-At (insloc 0 )))),k2) = insloc (card (ProgramPart (Directed J))) by A18, A51, SCMFSA8A:def 5
.= insloc (card (Directed J)) by AMI_1:105 ;
then n = card J by SCMFSA8A:34;
hence n + 2 < ((card I) + (card J)) + 3 by A52, NAT_1:13; :: thesis:

end;

suppose k2 < pseudo-LifeSpan s,(Directed J) ; :: thesis:

then insloc n in dom (Directed J) by A18, SCMFSA8A:31;
then n < card (Directed J) by SCMFSA6A:15;
then n + 2 < (card (Directed J)) + 2 by XREAL_1:8;
then A53: n + 2 < (card J) + 2 by SCMFSA8A:34;
((card I) + (card J)) + (1 + 2) = ((card J) + 2) + ((card I) + 1) ;
then (card J) + 2 <= ((card I) + (card J)) + 3 by NAT_1:11;
hence n + 2 < ((card I) + (card J)) + 3 by A53, XXREAL_0:2; :: thesis:

end;

end;

end;

then A54: insloc (n + 2) in dom (if=0 a,I,J) by Th56;
A55: IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = IC (Computation (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),2),k2) by A47, AMI_1:51
.= insloc (n + 2) by A1, A21, A12, A25, A28, A48, Th51 ;
CurInstr (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k) = (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) . (IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k)) by AMI_1:54
.= (if=0 a,I,J) . (IC (Computation (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))),k)) by A55, A54, Th26 ;
hence contradiction by A45, A55, A50, Th56; :: thesis:

end;

then LifeSpan (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = (pseudo-LifeSpan (s +* ((Directed J) +* (Start-At (insloc 0 )))),(Directed J)) + 3 by A32, A33, AMI_1:def 46;
hence LifeSpan (s +* ((if=0 a,I,J) +* (Start-At (insloc 0 )))) = (LifeSpan (s +* ((J ';' (Stop SCM+FSA )) +* (Start-At (insloc 0 ))))) + 3 by A18, A19, Th50; :: thesis: